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Theorem opelopabsb 4291
Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
opelopabsb  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Distinct variable groups:    x, y    x, B
Allowed substitution hints:    ph( x, y)    A( x, y)    B( y)

Proof of Theorem opelopabsb
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2804 . . . . . . . . . 10  |-  x  e. 
_V
2 vex 2804 . . . . . . . . . 10  |-  y  e. 
_V
31, 2opnzi 4259 . . . . . . . . 9  |-  <. x ,  y >.  =/=  (/)
4 simpl 443 . . . . . . . . . . 11  |-  ( (
(/)  =  <. x ,  y >.  /\  ph )  ->  (/)  =  <. x ,  y >. )
54eqcomd 2301 . . . . . . . . . 10  |-  ( (
(/)  =  <. x ,  y >.  /\  ph )  ->  <. x ,  y
>.  =  (/) )
65necon3ai 2499 . . . . . . . . 9  |-  ( <.
x ,  y >.  =/=  (/)  ->  -.  ( (/)  =  <. x ,  y
>.  /\  ph ) )
73, 6ax-mp 8 . . . . . . . 8  |-  -.  ( (/)  =  <. x ,  y
>.  /\  ph )
87nex 1545 . . . . . . 7  |-  -.  E. y ( (/)  =  <. x ,  y >.  /\  ph )
98nex 1545 . . . . . 6  |-  -.  E. x E. y ( (/)  =  <. x ,  y
>.  /\  ph )
10 elopab 4288 . . . . . 6  |-  ( (/)  e.  { <. x ,  y
>.  |  ph }  <->  E. x E. y ( (/)  =  <. x ,  y >.  /\  ph ) )
119, 10mtbir 290 . . . . 5  |-  -.  (/)  e.  { <. x ,  y >.  |  ph }
12 eleq1 2356 . . . . 5  |-  ( <. A ,  B >.  =  (/)  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  (/)  e.  { <. x ,  y >.  |  ph } ) )
1311, 12mtbiri 294 . . . 4  |-  ( <. A ,  B >.  =  (/)  ->  -.  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
1413necon2ai 2504 . . 3  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  <. A ,  B >.  =/=  (/) )
15 opnz 4258 . . 3  |-  ( <. A ,  B >.  =/=  (/) 
<->  ( A  e.  _V  /\  B  e.  _V )
)
1614, 15sylib 188 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  ( A  e.  _V  /\  B  e.  _V )
)
17 sbcex 3013 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  A  e.  _V )
18 spesbc 3085 . . . 4  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  E. x [. B  / 
y ]. ph )
19 sbcex 3013 . . . . 5  |-  ( [. B  /  y ]. ph  ->  B  e.  _V )
2019exlimiv 1624 . . . 4  |-  ( E. x [. B  / 
y ]. ph  ->  B  e.  _V )
2118, 20syl 15 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  B  e.  _V )
2217, 21jca 518 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  ( A  e.  _V  /\  B  e.  _V )
)
23 opeq1 3812 . . . . 5  |-  ( z  =  A  ->  <. z ,  w >.  =  <. A ,  w >. )
2423eleq1d 2362 . . . 4  |-  ( z  =  A  ->  ( <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. A ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
25 dfsbcq2 3007 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] [ w  /  y ] ph  <->  [. A  /  x ]. [ w  /  y ] ph ) )
2624, 25bibi12d 312 . . 3  |-  ( z  =  A  ->  (
( <. z ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ z  /  x ] [ w  /  y ] ph )  <->  ( <. A ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [. A  /  x ]. [ w  / 
y ] ph )
) )
27 opeq2 3813 . . . . 5  |-  ( w  =  B  ->  <. A ,  w >.  =  <. A ,  B >. )
2827eleq1d 2362 . . . 4  |-  ( w  =  B  ->  ( <. A ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } ) )
29 dfsbcq2 3007 . . . . 5  |-  ( w  =  B  ->  ( [ w  /  y ] ph  <->  [. B  /  y ]. ph ) )
3029sbcbidv 3058 . . . 4  |-  ( w  =  B  ->  ( [. A  /  x ]. [ w  /  y ] ph  <->  [. A  /  x ]. [. B  /  y ]. ph ) )
3128, 30bibi12d 312 . . 3  |-  ( w  =  B  ->  (
( <. A ,  w >.  e.  { <. x ,  y >.  |  ph } 
<-> 
[. A  /  x ]. [ w  /  y ] ph )  <->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph ) ) )
32 nfopab1 4101 . . . . . 6  |-  F/_ x { <. x ,  y
>.  |  ph }
3332nfel2 2444 . . . . 5  |-  F/ x <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }
34 nfs1v 2058 . . . . 5  |-  F/ x [ z  /  x ] [ w  /  y ] ph
3533, 34nfbi 1784 . . . 4  |-  F/ x
( <. z ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ z  /  x ] [ w  /  y ] ph )
36 opeq1 3812 . . . . . 6  |-  ( x  =  z  ->  <. x ,  w >.  =  <. z ,  w >. )
3736eleq1d 2362 . . . . 5  |-  ( x  =  z  ->  ( <. x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. z ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
38 sbequ12 1872 . . . . 5  |-  ( x  =  z  ->  ( [ w  /  y ] ph  <->  [ z  /  x ] [ w  /  y ] ph ) )
3937, 38bibi12d 312 . . . 4  |-  ( x  =  z  ->  (
( <. x ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ w  /  y ] ph )  <->  ( <. z ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ z  /  x ] [ w  /  y ] ph ) ) )
40 nfopab2 4102 . . . . . . 7  |-  F/_ y { <. x ,  y
>.  |  ph }
4140nfel2 2444 . . . . . 6  |-  F/ y
<. x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }
42 nfs1v 2058 . . . . . 6  |-  F/ y [ w  /  y ] ph
4341, 42nfbi 1784 . . . . 5  |-  F/ y ( <. x ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ w  /  y ] ph )
44 opeq2 3813 . . . . . . 7  |-  ( y  =  w  ->  <. x ,  y >.  =  <. x ,  w >. )
4544eleq1d 2362 . . . . . 6  |-  ( y  =  w  ->  ( <. x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  <. x ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
46 sbequ12 1872 . . . . . 6  |-  ( y  =  w  ->  ( ph 
<->  [ w  /  y ] ph ) )
4745, 46bibi12d 312 . . . . 5  |-  ( y  =  w  ->  (
( <. x ,  y
>.  e.  { <. x ,  y >.  |  ph } 
<-> 
ph )  <->  ( <. x ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ w  /  y ] ph ) ) )
48 opabid 4287 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
4943, 47, 48chvar 1939 . . . 4  |-  ( <.
x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] ph )
5035, 39, 49chvar 1939 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ z  /  x ] [ w  /  y ] ph )
5126, 31, 50vtocl2g 2860 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<-> 
[. A  /  x ]. [. B  /  y ]. ph ) )
5216, 22, 51pm5.21nii 342 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632   [wsb 1638    e. wcel 1696    =/= wne 2459   _Vcvv 2801   [.wsbc 3004   (/)c0 3468   <.cop 3656   {copab 4092
This theorem is referenced by:  brabsb  4292  opelopabaf  4304  opelopabf  4305  difopab  4833  isarep1  5347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094
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